Genderqueer is most commonly used to describe a person who feels that his/her gender identity does not fit into the socially constructed "norms" associated with his/her biological sex.
Genderqueer is an identity that falls anywhere between man/boy/male and woman/girl/female on the spectrum of gender identities.

Mary doesn't feel that s/he fits the mold of any one particular gender; therefore, Mary identifies as genderqueer.

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Any position in a wide variety of gender identities, spanning the spectrum between male and female. A person who is Genderqueer is not transgendered, though the option is open to transition. Being genderqueer has no bearing on sexual identity or orientation.

Eli might have a woman's body, but ze doesn't feel like a woman or a man. Eli is genderqueer.

The Urban Dictionary T-Shirt

"Genderqueer" describes a category of gender that is neither traditionally "female" or "male." Many different, nontraditional gender identities fall under the umbrella term "genderqueer." Genderqueer individuals may identify as having a gender somewhere between female and male or, more commonly, out of the gender binary all together.

Unless told otherwise, polite pronouns for genderqueer individuals are the singular they and ze/hir pronouns. Most genderqueer individuals take offense being referred to as "it."

"Is Dakota a girl or a boy?" "They're genderqueer!"

"So, because she's genderqueer, it means she's a slut and sleeps with everyone, right?" "Haha, no. Genderqueer just means their gender identity. It has nothing to do with their sexual orientation."

"I'm a dude, so I don't really want to take home ec." "Dude, I'm genderqueer! I'll take whatever the hell I want and not worry about it."

The Urban Dictionary T-Shirt

Gender Queer can be a term to describe a gender identity that is not encompassed by traditional roles of male and female.

This could be a person whose sense of self is the opposite of their perceived gender, such as a Transgendered person who is a “Male to Female” (T to F), or “Female to Male” (F to M).

Additionally, the term can be used for someone who is poorly described by traditional labels, and also not completely described by the term “Transgendered”. Thus, someone whose sense of self is not wholly male or female, but embodies elements of each gender could also be described as Gender Queer.

A Gender Queer person might respond to a question about their gender in many, many ways. Some might say, “There are male aspects to who I am, and female aspects to who I am. I value all those aspects of my sense of self and do not want to separate myself from the different parts of who I am.” Others might be crystal clear about their sense of self relating to gender, and find that the perception of society as to their gender is frequently wrong, and be totally okay with that. Still others may be on a journey to identifying their sense of self, and be perceived as one gender or another, sometimes correctly and other days not so much.

The overarching reality is that the way of identifying as either all male or female, or M to F or F to M, has has paved the way for the current level of comfort with gender being a more fluid, and less concrete concept.

The Urban Dictionary T-Shirt

Genderqueer people are people whose gender is not exclusively male or female. A similar term is nonbinary.

While genderqueer could be an identity of itself, some common genders that fall under the genderqueer umbrella are: agender, bigender, genderfluid, androgyne, ambigender, and neutrois.
Genderqueer is usually seen as falling under the trans umbrella, and some genderqueer people will call themselves trans.